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373 
P-/W4 


UC-NRLF 


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EXCHANGE 


THE  CRYSTELLIPTOMETER 


An  Instrument  for  the  Polariscopic  Analysis  of 
Very  Slender  Beams  of  Light 


BY 


LE  ROY  D.  WELD 


UNIVERST 

OF 


Submitted  in  Partial  Fulfillment  of  the  Requirements  for  the 
Degree  of  Doctor  of  Philosophy 


1922 
University  of  Iowa 


THE  CRYSTELLIPTOMETER 

AN  INSTRUMENT  FOR  THE  POLARISCOPIC  ANALYSIS  OF  VERY 
SLENDER  BEAMS  OF  LIGHT 

BY 

LE  ROY  D.  WELD 

I.  INTRODUCTORY 

The  determination  of  the  elements  of  elliptic  vibration  of  light, 
which  has  been  transformed  from  a  condition  of  plane  to  one  of 
elliptical  polarization,  is  found  to  reveal  so  much  concerning  the 
optical  nature  of  the  agencies  effecting  the  transformation  that  it 
becomes  a  matter  of  considerable  importance  to  be  able  to  make 
such  measurements  with  reasonable  accuracy.  It  so  happens  that 
the  ordinary  methods,  which  are  employed  for  the  elliptical  analy- 
sis, and  most  of  which  are  too  well  known  to  need  setting  forth 
here,  require  a  fair  breadth  of  field,  and  would  therefore  not  easily 
apply,  for  example,  to  a  ray  passing  through  a  pin-hole. 

The  research  presented  in  this  paper  had  its  origin  in  a  problem 
suggested  to  the  writer  some  years  ago  by  Dr.  L.  P.  Sieg,  and  was 
begun  in  1915.  It  has  been  carried  out  at  Coe  College  by  means 
of  apparatus,  much  of  which  was  kindly  loaned  for  the  purpose  by 
the  University  of  Iowa. 

Various  investigators  have  attempted  to  obtain  the  optical 
constants  of  metals  and  other  opaque  substances  by  reflection 
methods,  with  indifferent  success.  It  appears  certain  from  the 
research  of  Tate1  that  the  difficulty  has  been  due,  not  to  lack  of 
precision  in  the  analysis  of  the  elliptically  polarized  light  reflected 
from  the  opaque  surfaces,  but  to  the  variation  in  surface  condi- 
tions, brought  about  by  the  polishing  process,  which  modified 
the  nature  of  that  light.  Thus  it  was  found  impossible  to  prepare 
two  mirrors  of  the  same  kind  of  metal,  even  by  the  same  process, 
which  would  give  consistent  results,  while  different  measure- 
ments, and  even  different  kinds  of  reflection  measurements,  on 

1  Tate,  Phys.  Rev.,  34,  p.  321,  1912. 

67 


68  **         **w/WEW>        [J.O.S.A.  &  R.S.I.,  VI 

the  same  fresh  surface,  gave  good  agreement.  Dr.  Sieg's  suggestion 
was  that  opaque  crystals,  with  their  natural,  unpolished  facets, 
might  be  prepared  of  such  surface  purity  as  to  obviate  this  diffi- 
culty, and  at  the  same  time  furnish  material  for  an  interesting 
experimental  research  in  crystal  optics. 

Comparatively  few  large  opaque  crystals  can  be  obtained  in 
perfect  form.  Drude2  made  some  experiments  long  since  with  lead 
sulphide,  and  Muller3  subsequently,  with  antimony  sulphide, 
freshly  exposed  by  cleavage,  employing  ordinary  polariscopic 
methods.  In  1916  the  writer  made  a  preliminary  report4  on  the 
present  research,  which  had  already  given  unmistakable  evidence 
of  strong  double  refraction  in  minute  hexagonal  selenium  crystals, 
and  at  the  same  meeting,5  Dr.  Sieg  reported  having  found  by 
direct  photometric  measurements  that  the  selenium  crystal  has 
two  different  reflecting  powers  in  the  two  principal  directions.  Mr. 
C.  H.  Skinner  has  given  an  account6  of  his  interesting  observations 
on  selenium,  using  the  ordinary  Babinet  compensator  method, 
from  which  he  concluded  that  the  crystals,  like  artificially  prepared 
surfaces,  have  in  some  respects  their  individual  peculiarities,  espe- 
cially in  the  longitudinal  direction. 

The  method  adopted  by  the  writer  was  presented  to  the  Physical 
Society  in  1917  but  was  published  only  in  abstract7  at  that  time, 
the  research  being  then  still  in  progress.  It  is  applicable  not  only 
to  the  polariscopic  analysis  of  light  reflected  from  polished  plane 
surfaces  of  any  sort,  however  small  (such  as  minute  spikelets  or 
flakes  of  selenium,  tellurium,  cadmium,  etc.),  but  to  light  in  slen- 
der beams  from  any  source.  Its  application  has  recently  been 
suggested,  for  example,  to  the  comparison,  from  point  to  point, 
of  the  optical  properties  of  metal  film-deposits  of  non-uniform 
thickness.  Furthermore,  being  a  photographic  method,  it  is 
adaptable  to  the  ultra-violet,  and  the  apparatus  was  designed 
with  this  in  view.  Its  original  application  to  crystal-reflected, 

» Drude,  Ann.  d.  Physik,  36,  p.  532,  1889. 

3  Muller,  Neues  Jahrbuch  fur  Mineralogie,  17,  p.  187,  1903. 

4  Weld,  Proc.  Iowa  Acad.  Sci.,  1916,  p.  233. 

6  Sieg,  Proc.  Iowa  Acad.  Sci.,  1916,  p.  179. 
•  Skinner,  Phys.  Rev.,  N.S.  9,  p.  148,  1917. 

7  Weld,  Phys.  Rev.,  N.S.  11,  p.  249,  1918. 


Jan.,  1922] 


THE  CRYSTELLIPTOMETER 


69 


elliptically  polarized  light  is  what  suggested  to  the  writer  the  name 
"crystelliptometer"  for  the  distinctive  apparatus  employed. 

II.  PRINCIPLE  OF  THE  POLARISCOPIC  METHOD 

As  regards  the  polariscopic  analysis  itself,  the  method  employed 
is  a  modification  of  that  used  by  Voigt  for  the  study  of  polarized 
ultra-violet.  The  principle  is  stated  in  Voigt's  original  article8 
and  a  more  detailed  account  given  by  Mr.  R.  S.  Minor9  who 
applied  it  to  artificially  polished  mirrors  of  steel,  copper,  silver, 
etc.  A  non-mathematical  statement  of  the  original  method  will 
first  be  given. 

Let  us  have  a  uniform  parallel  beam  of  elliptically  polarized 
monochromatic  light;  to  determine  the  two  elements  of  the  elliptic 

a 

vibration,  which  may  be  taken  as  the  ratio  r  =  -of  the  X  to  the  Y 

o 

amplitude,  and  the  phase  advance  V  of  the  X  ahead  of  the  Y 
harmonic  component  (either  positive  or  negative).10 

R  C  Y 


=1 


\ 


\ 


Fig.  1 

Cross  hairs  and  quartz  wedge  systems.   XV ,  reticule.  C,  compensator.  R,  rotator 

This  beam  first  passes  through  a  pair  of  quartz  wedges  similar 
to  a  Babinet  compensator,  except  that  they  are  fixed  with  refer- 
ence to  each  other  (C,  Fig.  1).  The  edges  are,  let  us  say,  vertical, 
that  is,  parallel  to  the  Y  direction.  As  we  now  face  the  oncoming 

8  Voigt,  Physik.  Zeitschr.,  2,  p.  203,  1901. 

9  Minor,  Ann.  d.  Physik,  10,  p.  581,  1903. 

10  The  reason  for  here  using  the  inverted   V   instead  of  the  customary   A  will 
appear  later  (Sec.  VII). 


70 


LE  ROY  D.  WELD        [J.O.S.A.  &  R.S.I.,  VI 


light  and  move  across  the  emergent  beam  from  left  to  right,  we 
encounter  a  steady  increase  in  the  phase  advance,  so  that,  whereas 
the  light  through  the  center  of  the  compensator  has  phase 
difference  V  (the  same  as  the  original  light),  that  at  distance  x 
from  the  center  has  phase  difference  V+sx,  where  s  is  the  com- 
pensator constant  in  degrees  of  phase  per  millimeter.  Periodically, 
therefore,  we  shall  come  to  regions  of  light  plane  polarized  at  some 
angle,  viz.,  where  V  +sx  =  0,  TT,  2-Tr,  .... 

v 


-^     H 


/ 


\ 


Fig.  2 

Showing  action  of  rotator  on  a  strip  of  plane-polarized  light 

The  next  optical  member  is  another  pair  of  wedges,  which  will 
be  called  a  rotator  (R,  Fig.  1).  This  consists  of  a  wedge  D  of  right- 
handed,  and  one  L  of  left-handed,  quartz,  the  optic  axis  of  each 
being  along  the  path  of  the  light.  It  is  easily  seen  that  if  the 
light  entering  the  rotator  be  plane-polarized,  it  will  suffer  a  net 
rotation  of  plane  which  will  be  the  greater,  the  higher  the  point 
at  which  it  traverses  the  two  wedges.  This  rotation  may  be 
designated  by  qy,  q  being  the  rotator  constant  in  degrees  of  azi- 
muth per  millimeter. 

To  an  observer  facing  the  light  as  it  comes  from  the  rotator, 
any  one  of  the  periodic  vertical  ribbons  of  plane-polarized  light 
from  the  compensator  will  have  been  acted  upon  by  the  rotator  as 


Jan.,  1922]  THE  CRYSTELLIPTOMETER  71 

shown  in  Fig.  2.  Therefore  as  we  traverse  this  emergent  beam 
vertically  along  any  one  of  these  plane-polarized  regions,  we  shall 
encounter,  at  intervals,  light  vibrating  vertically,  and  half  way 
between  these,  light  vibrating  horizontally  (V  and  H,  Fig.  2)." 

Finally,  let  the  beam  be  passed  through  a  Nicol  set  so  as.  to 
extinguish,  say,  the  vertical  component.  Obviously,  wherever  we 
had  in  one  of  these  plane-polarized  strips  a  point  of  vertical 
vibration,  we  shall  now  have  a  black  spot.  The  field  will  then  be 
covered  with  black  spots  arranged  in  a  regular  pattern  of  vertical 
columns  and  horizontal  rows,  the  exact  design  of  which  will 
depend  upon  the  elements  of  the  original  elliptic  vibration.  Con- 
versely, these  elements  may  be  easily  deduced  from  the  observed 
arrangement  of  the  spot-pattern.  There  is  introduced  into  the 


Fig.  3 

Spot-pattern  for  elliptically  polarized  light  reflected  from  nickel.  (Enlarged.) 

field  a  pair  of  cross  hairs  corresponding  to  X  and  Y  axes  and 
coinciding  with  the  neutral  lines  of  rotator  and  compensator, 
respectively.  This  reticule  is  photographed  along  with  the  spot- 
pattern  and  the  coordinates  of  the  spots  determined  by  subsequent 
measurements  on  the  plate.  Such  a  spot-pattern,  with  the  cross 
hairs,  is  shown,  enlarged,  in  Fig.  3,  in  which  the  elliptic  polariza- 
tion was  produced  by  reflection  from  nickel. 

The  equations  relating  to  the  processes  just  described,  and 
the  practical  deduction  of  the  elliptic  elements  therefrom,  will  be 
worked  out  in  subsequent  sections  of  the  paper. 


72  LE  ROY  D.  WELD        [J.O.S.A.  &  R.S.I.,  VI 

III.  ADAPTATION  or  THE  METHOD  TO  SLENDER  BEAMS 

The  above  procedure,  as  followed  by  Voigt  and  others,  ob- 
viously requires  a  beam  of  light  whose  .cross  section  is  large  enough 
to  cover  the  entire  spot-pattern,  and  hence  would  not  be  adapted 
at  all,  for  example,  to  light  reflected  from  one  facet  of  a  small 
crystal.  The  writer's  modification  of  it  consists  simply  in  keeping 
the  very  slender  parallel  beam,  so  reflected,  stationary,  and  mov- 
ing the  whole  analyzing  system  of  quartz  wedges,  cross  hairs, 
Nicol  and  camera  back  and  forth  perpendicularly  across  it,  with 


AZIMUTK   CIRCCEr? 


Fig.  4  E 

Diagrammatic  layout  of  apparatus.    5,  source.    M,  monochromator.    Lt,  Ni,  collimator  and  polarizer. 

K,  reflecting  surface  mounted  in  front  of  drum  D.  X  to 

E,  crystelliptometer 

a  sort  of  weaving  motion,  until  the  whole  spot-pattern  has-been 
covered.  The  effect  is  the  same  as  if  the  beam  had  a  cross  section 
as  large  as  the  field  thus  traced  out.  Whenever  any  point  of  the 
analyzing  system  corresponding  to  one  of  the  dark  spots  arrives 
at  the  beam,  the  latter  is  extinguished  thereby,  and  the  spot  is 
left  on  the  plate.  In  practice  it  is  not  necessary  to  cover  the  whole 
pattern,  but  only  the  regions  of  it  in  which  the  rows  of  spots  are 
known  approximately  to  lie;  and  this  fact  saves  much  time. 


Jan,  1922] 


THE  CRYSTELLIPTOMETER 


73 


The  assemblage  of  optical  parts,  which  is  given  the  lateral 
motion  just  described,  together  with  its  mountings,  considered 
as  a  single  instrument,  is  what  has  been  referred  to  as  the 
"crys  tellip  tome  ter . " 

The  arrangement  of  the  writer's  apparatus  for  the  study  of 
crystals  is  shown  in  Fig.  4.  Light  from  the  source  S  is  focused 
by  the  quartz  lens  LI  on  the  inlet  slit  of  the  monochromator  M, 
from  whose  outlet  slit  the  monochromatic  light  diverges  and  is 
collimated  by  the  quartz  lens  L2.  (In  the  earlier  work,  color  filters 
were  used  instead  of  the  monochromator,  and  the  filtered  light 
focused  by  LI  directly  upon  the  slit  of  the  collimator.)  The 
parallel  beam  from  L2  is  plane-polarized  in  any  desired  azimuth 
by  Nij  a  large  polarizing  prism  with  square  ends  and  a  glycerine 
film. 


Fig.  5 

A,  spot-pattern  for  light  reflected  for  selenium  in  parallel  position.   B,  in  perpendicular 

position 

The  crystal  or  other  small  reflector  K  is  mounted  on  a  spectrom- 
eter, by  means  of  whose  telescope  T  and  verniers  V  any  desired 
angle  of  incidence  may  be  secured.  The  mounting  can  be  rotated 
so  that  it  is  very  easy  to  exhibit  the  double  refraction  of  selenium, 
tellurium,  etc,  the  spot-patterns  obtained  with  axis  horizontal 
and  with  axis  vertical  being  quite  visibly  different,  as  seen  in  Fig.  5. 

The  slender  reflected  beam,  in  general  elliptically  polarized, 
now  enters  the  analyzing  system  X,  C,  R,  N2  of  the  crystelliptom- 
eter.  X  represents  the  cross  hairs,  which  the  writer  has  found 
it  necessary  to  place  in  front  of  the  compensator  C.  (See  Sec.  VI.) 


74  LE  ROY  D.  WELD        [J.O.S.A.  &  R.S.I.,  VI 

The  compensator  wedges  have  an  angle  of  52',  which  gives  a 
relative  phase  change  of  about  144°  per  horizontal  millimeter  with 
sodium  light.  R  is  the  rotator,  with  wedge  angle  24°,  which  gives 
a  rotation  in  azimuth  of  about  16°  per  vertical  millimeter  with 
sodium  light.  The  dimensions  of  the  field  are  about  15  x  30  mm. 
The  wedge  system  and  cross  hairs  are  mounted  adjustably  in  a 
brass  tube  which  screws  upon  the  tube  containing  the  analyzing 
prism  Ni. 

Nz  is  a  duplicate  of  Ni,  the  two  being  interchangeable.  They 
are  rectangular  prisms  15  x  30  x  30  mm  consisting  of  Iceland  spar 
wedges  separated  by  glycerine,  the  angle  of  incidence  on  the  inter- 
face being  65°.  The  emergent  (extraordinary)  light  vibrates 
parallel  to  the  15  mm  dimension. 

The  spot-pattern  and  cross  hairs  are  photographed  together, 
by  means  of  the  quartz  lens  £3,  upon  the  plate  P.  Behind  the 
plate  is  an  eyepiece  E,  used  in  adjusting  the  focus  and  alignment 
before  the  plate  is  inserted.  The  plates  are  1  x  lj^  inch,  with 
emulsion  adapted  to  the  wave-length,  and  are  held  in  a  diminutive 
plate  holder. 

The  whole  crystelliptometer,  from  cross  hairs  X  to  eyepiece  E, 
is  contained  in  a  tube  about  85  cm  long,  so  mounted  on  two  pairs 
of  guides  at  right  angles  that  it  can  be  given  the  weaving  motion 
referred  to  in  order  to  trace  out  the  spot-pattern.  This  is  accom- 
plished by  means  of  two  micrometer  screws  perpendicular  to  each 
other.  One  of  these  screws  is  driven  by  a  worm-gear  electric 
motor  mechanism  so  as  to  move  the  instrument  slowly  across, 
along  a  line  determined  by  the  other  screw,  the  speed  varying 
with  the  exposure  required.  Furthermore,  the  whole  tube,  vith 
its  micrometer  mountings,  can  be  rotated  about  its  longitudinal 
axis  through  any  desired  angle,  thus  varying  the  relative  inclina- 
tion of  the  elliptic  vibration  to  be  analyzed,  with  respect  to  the 
coordinate  axes  of  the  analyzing  system.  The  extinction  plane  of 
the  analyzer  N2,  which  coincides  with  the  Y  axis,  being  first  verti- 
cal, one  spot-pattern  is  produced.  On  rotating  the  instrument 
through  an  angle  6  another  is  obtained,  and  so  on;  without,  how- 
ever, modifying  in  any  way  the  actual  nature  of  the  elliptic 
light  under  analysis.  The  equations  used  in  the  subsequent 


Jan.,  1922] 


THE  CRYSTELLIPTOMETER 


75 


theory  make  provision  for  this  arbitrary  angle  6,  the  advantage  of 
which  will  then  appear. 

The  source  of  light  at  first  employed  was  an  open  Nernst  fila- 
ment, which  has  later  been  replaced  by  a  special  low-voltage 
tungsten  ribbon  lamp.  In  some  preliminary  work  with  ultra- 
violet, an  iron  arc  was  employed. 

A  general  view  of  the  apparatus  is  shown  in  Fig.  6. 


Fig.  6 

General  view  of  crystelliptometer  and  accessory  apparatus 

The  spot-patterns  are  measured  upon  a  micrometer  comparator 
to  one-hundredth  of  a  millimeter.  The  lenses  were  removed  from 
the  microscope,  and  a  pin-hole  and  hair-line  substituted,  a  device 
suggested  to  the  writer  by  Dr.  Elmer  Dershem.  Not  a  little  of 
the  routine  work  consists  of  the  p'ate  measurements  and  their 
reduction.  It  has  been  found  possible,  with  good,  clear  plates,  to 
locate  a  spot  by  a  single  measurement  with  a  probable  error  of 
less  than  one-hundredth  of  a  millimeter.  A  selenium  spot-pattern 
and  the  corresponding  "comparison  plate"  (see  Sec. VI)  are  shown 


76 


LE  ROY  D.  WELD        [J.O.S.A.  &  R.S.I.,  VI 


together  as  A,  B  in  Fig.  7,  and  another  similar  pair,  with  greater 
wave-length,  as  C,  D. 

The  quartz  lenses  and  wedges  and  the  Iceland  spar  wedges  for 
the  Nicols,  designed  by  the  writer,  were  made  by  Hilger  of  London, 
as  was  also  the  monochromator;  the  comparator  by  Gaertner  of 
Chicago.  The  remainder  of  the  special  apparatus,  including  the 
driving  mechanism,  was  built  at  the  University  of  Iowa  by  Messrs. 
M.  H.  Teeuwen  and  J.  B.  Dempster,  assisted  by  the  writer. 


Fig.  7 

A,  selenium  spot-pattern.    B,  corresponding  comparison  plate.    C,  D,  same,  with  greater  wave-length. 
Note  the  greater  spot-intervals 

IV.  OUTLINE  OF  THE  MATHEMATICAL  THEORY  OF  THE 
ANALYZING  SYSTEM 

The  mathematical  theory  of  the  action  of  the  analyzing  system, 
as  used  in  the  writer's  method,  and  of  the  determination  of  the 
elliptic  elements,  will  now  be  briefly  given.11  In  the  notation  here 
used,  X  and  Y  are  components  of  the  vibration  displacement  of 
the  light,  while  x  and  y  are  coordinates  of  points  of  the  field  with 
reference  to  the  axes. 


11  This  is  necessary  for  the  reason  that  the  present  method  departs  in  certain  essen- 
tial particulars  from  the  original  and  gives  rise  to  a  different  set  of  equations.  Minor 
did  not,  for  example,  rotate  the  quartz  wedge  system  through  the  arbitrary  angle  6 
which  makes  possible  the  least  square  adjustment  of  the  observations,  and  which, 
as  will  be  easily  seen,  also  provides  automatically  for  any  difficulty  due  to  the  azimuth 
of  the  light  under  examination  happening  to  be  very  small,  without  altering  that 
light  in  any  way. 


Jan.,  1922]  THE  CRYSTELLIPTOMETER  77 

Let  the  harmonic  components  of  the  elliptic  vibration  to  be  an- 
alyzed, as  viewed  by  one  facing  the  oncoming  waves,  be  given  by 
the  equations 

X  =  a  cos  (cot-fV) (1) 

F  =  b  cos  cot (2) 

If  the  analyzing  system  by  now  rotated  through  an  angle +6, 
these  equations  are  transformed,  with  reference  to  the  new  axes, 
into 

X'  =  a  cos  B  cos  (cot +V)  +b  sin  0  cos  cot. (3) 

F'  =  b  cos  6  cos  cot-  a  sin  8  cos  (cot+V) (4) 

Upon  passage  through  the  compensator,  the  F-component 
receives  an  advance  of  phase,  varying  with  the  abscissa  x  of  the 
point  where  it  passes  through,  and  equal  to  sx.  (3)  and  (4)  then 
become 

X"  =  a  cos  0  cos  (cot+V)  +b  sin  6  cos  cot, .  .  . (5) 

F"  =  6fcos  0cos  (tot+sx)-a  sin  6  cos  (cot+V  +sx)(6) 
Upon  passage  through  the  rotator,  there  is  a  simple  rotation  of 
the  axes  of  the  ellipse,  without  further  change,  through  an  angle 
qy,  which  corresponds  to  a  rotation  —  qy  of  the  coordinate  axes,  so 
that  the  components  of  the  finally  emergent  light  are 
X'"  =  a  cos  6  cos  qy  cos  (cot+V)  +6  sin  6  cos  673;  cos  cot 

—  b  cos  6  sin  qy  cos  (cot  +sx)+a  sin  6  sin  qy  cos  (cot+V  +sx). 

_ (V) 

Yf"  =  (a  corresponding  long  expression  which  we  shall  not 

need) (8) 

The  light  now  passes  through  the  Nicol  N2,  which  shuts  off 
the  F-component  all  over  the  field;  hence  (8)  is  not  needed. 
Only  X'"  gets  through,  and  this  will  vanish,  leaving  the  dark 
spots,  at  every  point  where  x  and  y  have  such  values  as  to  render 
the  expression  (7)  equal  to  zero  all  the  time,  that  is,  independently 
of  the  value  of  t.  To  fulfill  this  condition,  dividing  (7)  by  b  cos  6, 

letting  -  =  r,  expanding  the  parentheses  containing  co/  and  group- 
b 

ing  the  terms  in  sin  cot  and  cos  cot  separately,  we  have 
[  —  r  tan  6  sin  qy  sin  (V  +sx)  +sin  qy  sin  sx 

—  r  cos  qy  sin  V]  sin  cot 


78  LE  ROY  D.  WELD       [J.O.S.A.  &  R.S.I.,  VI 

+[tan  0  cos  qy+ r  cos  qy  cos  V  —sin  qy  cos  sx 

-f-r  tan  9  sin  qy  cos  (V+sx)]  cos  cot  =  0. 

That  this  may  be  true  independently  of  /,  the  coefficients  must 
separately  vanish,  giving 

—  r  tan  6  sin  qy  sin  (V  +sx)  -f  sin  qy  sin  sx  —  r  cos  qy  sin  V  =  0 .  .  (9) 
tan  6  [cos  qy-\-r  sin  qy  cos  (V+s#)]  -\-r  cos  g;y  cos  V  —  sin  qy  cos  sx 

=  0 (10) 

Letting  tan  6  =  m,  expanding  the  functions  of  V+stf,  and  collect- 
ing, these  become 
[m  sin  qy  cos  sx + cos  qy]r  sin  V  -f-  m  sin  qy  sin  s#.  r  cos  V 

=  sin  qy  sin  s# (11) 

m  sin  £y  sin  $#.  r  sin  V  -f  [m  sin  g;y  cos  sx+ cos  g;y]r  cos  V 

=smqycossx—mcosqy (12) 

In  these  equations,  x  and  y  are  the  measured  coordinates  of 
any  dark  spot,  5  and  q  are  the  compensator  and  rotator  constants, 
determined  from  the  spot  intervals,  and  m  is  the  tangent  of  the 
known  angle  6.  Hence  everything  is  known  except  r  and  V. 
Letting 

m  sin  qy  cos  s#+cos  qy  =  H, 
m  sin  qy  sin  sx  =  K, 

sin  qy  sin  sx  =L, 

sin  qy  cos  sx  —  m  cos  qy  =  P, 
(11)  and  (12)  become 

H-r  smV+K-r  cos  V  =L (14) 

K-r  smV+H-r  cos  V  =P (15) 

the  solution  of  which,  as  simultaneous  equations,  gives  the  re- 
quired elliptic  elements 


(13) 


LH-PK 

tan  V  = 

PH-LK 

It  is  thus  theoretically  possible  to  deduce  the  elements  from 
the  measured  coordinates  of  any  single  spot. 


Jan.,  1922] 


THE  CRYSTELLIPTOMETER 


79 


V.  APPLICATION  TO  SPOT-PATTERNS 

If  we  place  under  examination  light  which  is  plane-polarized 
at  azimuth  45°,  so  that  r  =  1  and  V  =  0,  it  will  be  seen  from  the 
manner  of  their  formation,  as  described  in  Sec.  II,  that  the  spots 
will  be  symmetrically  arranged  with  respect  to  the  F-axis  in 
equally  spaced  horizontal  rows.  This  is  the  condition  of  things 
on  the  comparison  plate,  Fig.  7  B  (see  Sec.  VI).  Let  the  distance 
apart  of  the  spots  in  the  rows  be  d,  and  in  the  vertical  columns, 

(Fig.  8).  These  are  easily  measured  (see  Sec.  VI).  The  com- 
pensator constant  s  is  equal  to ,  and  the  rotator  constant  q 

d 

1 80 

equals  — ,  in  degrees  per  millimeter  (see  Sec.  II). 
o 


\ 

PH      •    +* 

< 

k 

III! 

1      1      •+' 

—  4      I      t 

I      •      I    -• 

•      I      t      1 

•      I      I-2 

i    t    » 

r    *    t    t* 

444 

s 

441- 

itt 

r    t    t    t  - 

444 

444* 

Fig.  8 

Diagram  of  spots  on  a  comparison  plate,  taken 
with  plane-polarized  light  at  45°  azimuth 


Fig.  9 

Diagram  of  spots  on  a  pattern  taken  with  light 

elliptically  polarized  at  phase  difference 

202°  and  azimuth  29° 


If  now  we  introduce  by  any  means  a  phase  difference  V  between 
the  X-  and  F-components,  there  will  be  a  uniform  lateral  dis- 
placement a  of  the  whole  spot  system,  so  that  the  coordinates  of 

the  spot  5,  which  in  Fig.  8  are  0,  — ,  now  become  a,  — .  Then,  again, 

if  a  change  is  produced  in  the  azimuth  of  the  incident  light,  the 
alternate  rows  +1,  —2,  etc.  will  be  displaced  vertically  one  way, 


80  LE  ROY  D.  WELD        [J.O.S.A.  &  R.S.I.,  VI 

and  rows  +2,  —1,  etc.,  the  other  way,  through  a  certain  amount 
a,  so  that  now  the  coordinates  of  S  are  x  —  a,  y  =  —  -fa  (a  being  neg- 
ative). These  changes  are  shown  in  Fig.  9.12 

Introducing  the  above  values  of  s,  q,  x,  and  y  into  (13),   and 
resuming  m  =  tan  0,  we  get 

[~\      r         T          r~  n 

45°+ -180°   cos  -360°    +cos  45° -f  -180° 
5        J      La       J          L          8        _r 

[a         "1         fa         ~| 
45°  +  -  180°   sin     -360°   , 

1      Fa       T  '(18) 

L   =  sin    45°  +  -  180°  sin  ^360°   , 
L  5         J      Id        J 

P  =  sin|450H-^  180°  |cosfo60°] -tan  (9  cos|45°H-- 180°  1. 
L  d         J       La         J  L  S         J 

While  the  attention  is,  indeed,  fixed  on  one  particular  spot  5, 

5 
Fig.  9,  whose  coordinates  are  a  and  —  -fa,  yet  in  actual  practice  it 

is  expedient  to  make  measurements  on  a  number  of  spots,  usually 
twenty  or  more,  and  deduce  the  position  of  S  from  them.  Refer- 
ring to  Fig.  9,  it  is  clear  that  the  abscissa  of  any  spot  in  the  nth 
vertical  column  (numbered  along  the  bottom  of  the  figure)  is 

d 


(19) 


which  gives 

n 


By  measuring  the  abscissas  of  several  spots  in  different  rows, 
thus  varying  x  and  n,  we  obta'n  as  many  independent  observa- 
tions upon  a. 

17  If  the  azimuth  is  made  90°,  rows  +1,  +2,  rows  —1,  —2,  etc.,  will  merge  or 
dove-tail  together  in  pairs  forming  continuous  horizontal  dark  stripes.  If  it  is  made  0, 
the  pairs  of  rows  +1,  —  1,  etc.,  will  similarly  coincide.  This  affords  a  good  means  of 
adjusting  the  quartz  wedge  system  with  respect  to  the  previously  adjusted  analyzing 
Nicol  (see  Sec.  VI). 


Jan.,  1922]  THE  CRYSTELLIPTOMETER  81 

Again,  the  ordinate  of  any  spot  in  the   *>th  horizontal  row 
(counted  upward  or  downward  from  the  X-axis)  may  be  seen  to  be 


or         .-±(-1)'-   '       +  j..  ..(20) 


(±  according  as  the  row  is  above  or  below  the  X-axis),  and  we 
shall  have,  therefore,  as  many  observations  upon  a  as  there  are 
spots  measured. 

The  averages  from  these  observations  on  a  and  a  are  easily 
obtained  by  means  of  formulas  depending  upon  the  particular 
selection  of  spots  made.  If  the  selection  consists  of  an  equal  num- 
ber of  vertical  columns  on  each  side  of  the  F-axis  and  of  horizontal 
rows  above  and  below  the  X-axis  (as  should  be  the  case  for  other 
reasons),  these  averaging  formulas  become  quite  simple,  a,  a,  d 
and  5  being  thus  determined  from  the  measurements  on  the  plate, 
substitution  of  their  values  in  (18)  gives  the  necessary  constants 
H,  K,  L,  P,  appearing  in  the  expressions  for  r  and  tan  V,  Eqs. 
(16),  (17),  and  the  problem  is  solved  so  far  as  is  possible  from  a  sin- 
gle experiment. 

We  may,  however,  expose  other  plates  with  different  values  of 
0,  obtained  by  rotating  the  crystelliptometer  tube  about  its  own 
axis.  This  does  not  alter  the  elliptic  elements,  but  it  gives  new 
values  of  H,  K,  L,  P.  In  such  case,  (14)  and  (15)  may  be  used  as 
observation  equations  of  the  first  degree  with  r  sin  V  and  r  cos  V 
as  unknowns,  and  we  may  obtain  as  many  different  pairs  of  them 
as  there  are  measured  plates,  finally  adjusting  them  by  the  method 
of  least  squares.  It  has  been  the  writer's  practice  to  assign  a 
weight  to  each  measured  plate  by  means  of  the  grading  method,13 
each  plate  characteristic,  such  as  clearness,  symmetry  of  spots, 
etc.,  being  graded  separately. 

It  should  be  stated  that,  except  in  cases  where  the  number  of  spots 
on  a  plate  available  for  measurement  is  too  limited  for  precision,  a 

13  Weld,  A  Method  of  Assigning  Weights  to  Original  Observations,  Science,   50, 
p.  461,  1919. 


82  LE  ROY  D.  WELD        [J.O.S.A.  &  R.S.I.,  VI 

single  plate,  with  6  =  0,  is  sufficient,  and  a  vast  amount  of  labo- 
rious calculation  is  thus  avoided.  For  with  6  =  0  in  (18),  the  elliptic 
elements,  given  by  (16)  and  (17),  reduce  at  once  to 

(21) 


(22) 


It  is  only  with  light  of  very  large  wave-length  that  the  number 
of  spots  in  the  field  is  likely  to  be  so  few  as  to  require  the  repeated 
exposures. 

VI.  MISCELLANEOUS  DETAILS  AND  SOURCES  OF  ERROR 

The  general  arrangement  of  the  apparatus  as  employed  for 
the  study  of  opaque  crystals  was  explained  in  Sec.  III.  In  order 
to  adjust  all  the  parts  in  proper  relation  to  each  other,  use  is 
made  of  a  sensitive  cathetometer  set  with  reference  to  the  pier 
on  which  the  apparatus  stands.  By  this  means  is  secured  the 
horizontality  of  the  monochromator,  the  collimator,  the  polarized 
incident  beam,  the  reflected  beam  from  the  crystal,  and  the 
cry  stellip  tome  ter  tube,  taken  in  the  order  named.  The  plane  of 
incidence  and  reflection  is  thus  strictly  horizontal,  and  the  azi- 
muth of  polarization,  the  arbitrary  angle  6,  and  the  vibration  com- 
ponents X  and  Y  are  reckoned  with  reference  to  it.  The  crystal 
is  mounted  on  the  end  of  a  small  rod  in  front  of  the  dark  opening 
into  a  hollow  drum,  —  a  black  body,  so  to  speak,  which  makes  a 
perfectly  dead  back-ground.  The  mounting  may  be  turned  in  a 
vertical  plane,  and  is  provided  with  a  graduated  circle,  so  as  to 
give  the  crystal  any  desired  angle  from  0°  to  180°  with  the  plane 
of  reflection.  The  drum  (D,  Fig.  4)  is  placed  on  a  spectrometer 
prism  table  for  adjusting  the  angle  of  incidence,  as  previously 
explained. 

Considerable  trouble  has  been  experienced  with  the  mono- 
chromator, inasmuch  as  no  reliance  can  apparently  be  placed 
upon  the  wave-lengths  indicated  by  it;  and  furthermore,  the  wave- 
length corresponding  to  any  given  setting  is  found  to  vary  from 
day  to  day.  In  all  final  work  it  has  therefore  been  customary  to 


Jan.,  1922]  THE  CRYSTELLIPTOMETER  83 

divert  the  light  emerging  from  the  monochromator  into  a  separate 
grating  spectrometer  (not  shown  in  Fig.  4)  and  compare  it  with 
the  sodium  standard  just  before  making  each  exposure.  Another 
and  more  serious  difficulty  with  the  monochromator  is  the  impur- 
ity of  the  light  furnished,  especially  in  the  shorter  wave-lengths. 

The  accurate  adjustment  of  the  focus  of  the  collimating  and 
camera  lenses  is  a  matter  of  some  importance,  especially  the 
latter.  These  quartz  lenses  are,  of  course,  not  achromatic,  and 
the  focus  must  be  calibrated  for  wave-length.  In  the  case  of  the 
collimator,  it  has  sufficed  to  measure  the  focal  length  for  one 
wave-length  on  an  optical  bench  and  calibrate  the  tube  from  the 
known  dispersion  of  quartz.  But  any  inaccuracy  in  the  focus 
of  the  camera  lens  will  result  in  displacements  of  the  spot  images 
and  resultant  errors  in  the  elliptic  elements,  so  that  this  requires 
greater  precision.  The  method  here  employed  is  one  devised  by 
the  writer  and  referred  to  as  the  "offset"  or  "broken  prism" 
method.14  The  proper  focus  for  a  given  wave-length  may  be  thus 
obtained  with  a  probable  error  of  only  one  or  two  tenths  of  a 
millimeter,  and  it  is  easy  to  calibrate  the  focus  tube  accordingly. 

The  need  for  a  special  precaution  arises  from  the  fact  that 
the  spot-pattern  in  the  crystelliptometer  appears  to  be  a  sort  of 
virtual  image  lying  in  a  definite  plane.  It  is  necessary  to  get 
the  cross  hairs  accurately  into  that  plane,  otherwise  there  will 
be  an  apparent  parallax  between  cross  hairs  and  spots,  and  the 
results  will  be  seriously  affected  if  the  light  happens  to  be  not 
strictly  parallel  to  the  crystelliptometer  axis.  Furthermore,  the 
position  of  this  virtual  plane  is  found  to  vary  systematically 
with  the  wave-length,  so  that  the  adjustment  has  to  be  made  for 
each  wave-length  used.  This  is  accomplished  by  mounting  the 
cross  hairs  in  a  ring  having  a  longitudinal  micrometer  movement 
in  .the  tube.  The  crystelliptometer  is  turned  a  little  to  right 
and  left  with  respect  to  the  beam  of  light  and  the  reticule  moved 
forward  or  backward  until  the  parallax  disappears.  It  has  always 
been  found  necessary,  in  the  visible  spectrum,  to  place  the  reticule 
in  front  of  the  compensator,  as  in  Fig.  4. 

14  Weld,  Some  Precise  Methods  of  Focusing  Lenses,  School  Science  and  Mathe- 
matics, 18,  p.  547,  1918. 


84  LE  ROY  D.  WELD        [J.O.S.A.  &  R.S.I.,  VI 

The  cross  hairs  in  the  writer's  instrument  are  of  very  fine 
spun  glass,  their  images  representing  on  the  plate  the  X-  and  F- 
axes  of  the  spot-pattern.  The  ring  in  which  they  are  mounted  is 
provided  with  the  usual  lateral  adjusting  screws.  It  is  inevitable 
that  the  reticule  will  get  out  of  adjustment  laterally,  and  for  the 
purpose  of  determining  this  error  (which  would  be  serious  if  neg- 
lected), what  are  called  comparison  plates  are  taken  at  frequent 
intervals,  using  a  full-sized  beam  of  parallel,  strictly  plane- 
polarized  light  direct  from  the  polarizing  Nicol.  The  cross  hairs 
are  first  given  approximate  adjustment  visually,  using  this  light. 
The  comparison  plate  is  then  taken  and  the  small  errors  of  cross 
hair  adjustment  remaining,  amounting  to  a  few  thousandths  of  a 
millimeter,  are  determined  by  measurements  upon  it  and  proper 
allowance  made  for  them  in  the  reduction  of  other  plates.  If  a 
symmetrical  pattern  is  selected,  the  adjustment  of  the  horizontal 
cross  hair  is  really  immaterial,  as  the  true  Jf-axis  can  readily  be 
found  as  the  mean  position  of  the  horizontal  rows  of  spots  em- 
ployed in  measuring  any  plate. 

Owing,  no  doubt,  to  slight  inequality  of  the  quartz  wedge 
angles,  the  cross  hair  adjustment  error  is  found,  like  the  parallax, 
to  vary  systematically  with  the  wave-length,  and  a  new  compari- 
son plate  must  therefore  be  taken  for  each  wave-length  used.  It 
is  very  difficult,  even  with  the  spectrometer  test,  to  keep  the  wave- 
length strictly  constant  through  a  series  of  experiments.  Curiously 
enough,  the  most  sensitive,  and  the  final,  check  on  wave-length 
has  been  found  to  be  the  spot-interval  d  (Figs.  8,  9),  which  can 
be  measured  with  great  precision  (see  below).  The  greater  the 
wave-length,  the  farther  apart  are  the  spots,  both  horizontally 
and  vertically,  on  account  of  the  dependence  of  both  the  phase- 
relation  change,  and  the  rotation,  in  quartz,  upon  the  wave- 
length. (See  Fig.  7.)  In  practice,  it  is  found  advisable  to 
determine,  by  means  of  auxiliary  comparison  plates,  the  relation 
of  the  cross  hair  error  to  d  for  light  in  each  region  of  the  spectrum 
used,  and  to  deduce  the  required  correction  from  the  measured 
d  on  each  elliptic  plate. 

The  accurate  determination  of  the  spot  intervals  d  and  6  for 
each  plate  is  an  essentially  vital  part  of  the  work.  The  most 


Jan.,  1922]  THE  CRYSTELLIPTOMETER  85 

probable  value  of  d  can  be  deduced  fr  m  the  plate  measurements 
by  forming  observation  equations  from  the  abscissas  of  the  spots 
taken  by  groups  in  each  row,  and  applying  the  simple  least  square 
adjustment  requisite  to  the  case.  This  is  all  done  very  quickly  by 
means  of  a  formula  which  is  the  same  for  all  spot-patterns  similarly 
selected.  With  a  symmetrical'spot-pattern 5 is  simply  twice  the 
mean  absolute  ordinate  of  the  spots  (without  sign).  Thus  no 
extra  measurements  are  necessary  for  d  and  8.  But  it  will  not  do 
to  rely  on  the  comparison  plate,  or  any  other  one  plate,  for  the 
spot-intervals  corresponding  to  a  supposedly  fixed  wave-length, 
as  variations  of  wave-length  too  slight  to  have  noticeable  effect  on 
the  ordinary  properties  of  the  light,  will  cause  serious  errors 
through  this  means. 

When,  as  usual,  the  arbitrary  orientation  (Sec.  IV)  is  zero,  it  is 
seen  by  Eq.  (21)  that  the  ratio  r  of  the  two  vibration  components 
of  the  unknown  light,  which  is  the  tangent  of  the  azimuth,  is  given 
in  terms  of  a  and  5.  These  are  determined  from  the  y  measure- 
ments alone.  For  some  reason  not  certainly  explained,  it  has  been 
found  that  there  is  a  persistent  error  in  r  as  deduced  from  the 
measurements  on  the  spot-pattern,  whose  value  appears  to  be  a 
linear  function  of  r  itself.  The  error  is  eliminated  by  first  finding 
the  uncorrected  value  of  r  from  the  plate  measurements,  and  then 
taking  a  correction  pattern  with  plane-polarized  light  having 
azimuth  set  for  that  value  of  r  (which  is  thus  accurately  known). 
The  measurement  of  this  plate,  and  the  comparison  of  the  erro- 
neous value  of  r  deduced  therefrom  with  the  true  value  given  by 
the  polarizer  azimuth  circle,  afford  the  necessary  correction, 
which  sometimes  amounts  to  as  much  as  two  or  three  degrees. 
This  must  be  done  for  each  wave-length  used.  These  azimuth- 
correction  plates,  like  the  comparison  plates,  are  taken  with  full- 
sized  beam  from  the  polarizer  and  require  only  a  short  exposure. 

One  more  detail  of  technique  is  worthy  of  note.  It  will  be 
noticed  from  Fig.  7C,  for  example,  that  on  some  patterns  the  spots 
are  not  symmetrical  vertically,  but  are  decidedly  triangular  or 
cuneiform,  and  tend  to  run  together  in  double  rows.  This  makes 
it  difficult  to  estimate  the  Y  position  of  the  spot  nucleus  with 
certainty  in  measuring  these  plates.  It  will  not  do  to  bisect  the 


86 


LE  ROY  D.  WELD        [J.O.S.A.  &  R.S.I.,  VI 


spot  with  the  micrometer  in  this  direction.  In  order  to  ascertain 
the  location  of  the  nucleus  within  the  spot,  the  means  adopted 
has  been  to  calculate,  theoretically,  the  geometrical  form  of  the 
concentric  lines  of  equal  intensity  surrounding  it,  taking  the  con- 
stants from  actual  measurements  on  typical  plates.  The  form 
of  these  lines,  derived  from  an  application  of  the  mean  value 
theorem,  is 

cos(7-^)  =  1-J/-(1-r2)c0s2(?y..  ..(23) 

r  sin  2qy 

in  which  If  is  a  parameter  depending  on  the  intensity  along  the 
curve  in  question;  for  the  nucleus,  M=0.  x  and  y  are  the  coordinate 


Fig.  10 

Lines  of  equal  intensity  about  the  spots  of  Fig.  5B,  greatly  enlarged 

variables,  and  the  other  quantities  have  the  same  meanings  as  in 
Sec.  IV.  Fig.  10  shows  the  nuclei  and  curves  corresponding  to 
M  =  0,  0.25  and  0.50  on  the  plate  shown  in  Fig.  5B.  With  the 
knowledge  afforded  by  such  a  figure,  it  is  easy  to  estimate  with 
some  accuracy  what  point  within  the  spot  is  to  be  aimed  at  in 
making  the  measurement.  The  nucleus  appears  to  represent 
approximately  the  center  of  gravity  of  the  spot  area,  rather  than 
to  bisect  its  vertical  dimension. 


Jan.,  1922]  THE  CRYSTELLIPTOMETER  87 

VII.     TYPICAL  APPLICATIONS.  REFLECTION  FROM  SELENIUM 

CRYSTALS 

Among  the  preliminary  tests  of  the  cry stellip tome ter  and  the 
methods  of  using  it  set  forth  in  the  foregoing  sections  were  the 
analysis  of  polarized  light  rendered  elliptical  by  reflection  from 
nickel  and  copper  mirrors  or  from  crystals  of  lead  sulphide  and 
tellurium,  or  by  passage  through  sheets  of  mica;  and  a  tryout 
of  FresnePs  equations  for  the  rotation  of  plane-polarized  light 
reflected  from  glass.  The  only  problem,  however,  upon  which 
serious  attack  has  yet  been  made  by  the  crystelliptometer  method 
is  the  experimental  part  of  an  extensive  research  now  in  progress 
at  the  University  of  Iowa,  viz.,  the  optical  laws  of  absorbing 
crystals.  The  general  theory  of  this  subject  was  handled  with 
great  thoroughness  by  Drude  in  his  inaugural  dissertation15  many 
years  ago,  but  until  recently  the  only  experimental  data  upon 
which  tests  of  the  theory  might  be  based  have  been  with  reference 
to  certain  large  crystals  of  the  rhombic  system.  The  unique  elec- 
tro-optical properties  of  the  hexagonal  selenium  crystal  have  sug- 
gested a  revival  of  the  subject,  the  outgrowth  of  which  is  the 
research  in  question.  The  few  data  given  below  are  the  first  final 
results  obtained  by  this  method,  and  will  serve  to  illustrate 
it.  They  are  summarized  from  a  long  series  of  measurements  on 
many  selenium  crystals,  and  involved  the  taking  of  nearly  three 
hundred  plates.  The  instrument-  is  now  in  the  hands  of  other 
observers  whose  aim  is  to  accumulate  information  regarding 
the  optical  properties  of  small  crystals  of  various  metallic  sub- 
stances. 

Some  of  the  finest  crystals  of  hexagonal  selenium  ever  prepared 
were  kindly  put  at  the  writer's  disposal  by  Dr.  E.  0.  Dieterich, 
who  produced  them  by  sublimation  at  the  University  of  Iowa. 
They  are  2  or  3  cm  long  and  with  facets  often  0 . 5  mm  in  width. 
From  some  hundreds  of  these,  about  a  dozen  superb  specimens 
were  selected  and  mounted  for  use  with  the  crystelliptometer. 
Selenium  has  a  tendency  to  twist  and  warp,  and  great  care  had  to 
be  exercised  to  select  crystals  with  plane  facets.  Even  with  these  it 

16  Drude,  Ann.  d.  Physik,  32,  p.  584,  1887. 


88 


LE  ROY  D.  WELD        [J.O.S.A.  &  R.S.I.,  VI 


was  usually  found  expedient  to  limit  the  illumination  to  only  one 
or  two  millimeters  of  length,  so  that  an  area  of  one  square  milli- 
meter or  less  of  reflecting  surface  was  quite  typical. 

Much  of  the  work  was  carried  on  at  wave-lengths  near  the  mid- 
dle of  the  visible  spectrum.  The  incident  light  was,  in  every  case, 
plane-polarized  at  azimuth  45°.  Several  crystals  were  tested 
in  both  horizontal  and  vertical  positions  at  wave-length  0.5/z, 
and  at  incidence  angles  45°  and  60°.  At  the  other  wave-lengths  the 
incidence  angle  was  maintained  at  60°.  The  wave-lengths  prin- 
cipally used  were  0.45/z,  0.50 /*,  0.55 /*,  0.65 ju,  and  0.70;*,  a  range 
sufficient  to  give  typical  results.  The  wave-length  0.60/z  was  de- 
ferred to  a  separate  research,  for  the  reason  that  the  data  obtained 
by  C.  H.  Skinner16  and  others  with  selenium  contain  certain  anoma- 
lies near  this  point,  while  the  writer's  results  are  strongly  sugges- 
tive of  what,  with  a  transparent  substance,  would  correspond  to  an 
absorption  band,  in  the  neighborhood  of  this  wave-length.  The 
matter  deserves,  therefore,  more  minute  investigation. 

___A_=2_e_°_     . 


Fig.  11 

A,  incident  vibrations,  plane-polarized.  B,  C,  elliptic  vibrations  viewed  looking  with  and  against  the  beam, 

respectively 

Upon  experimenting  with  a  number  of  crystals  prepared  at 

different  times,  it  was  found  that,  with  one  or  two  exceptions,  they 

gave  fairly  consistent  results,  though  all  the  crystals  were  several 

months  old  when  the  experiments  were  begun.     Great  care,  however, 

16  Loc.  dt. 


Jan.,  1922] 


THE  CRYSTELLIPTOMETER 


89 


had  been  taken  to  keep  them  clean  and  away  from  contact 
with  fumes  or  corrosive  gases.  It  is  quite  possible  that  these 
exceptions  were  due  to  surfaces  that  had  in  some  way  become 
tarnished  or  contaminated,  in  spite  of  the  precautions.  After 
the  first  trials,  three  crystals  were  selected  which  gave  the  clearest 
spot-patterns,  and  subsequent  work  was  confined  to  these.  The 
data  in  the  accompanying  tables  are  the  weighted  means  of  the 
results  obtained  at  the  respective  wave-lengths.  It  should  be 
stated  that  those  corresponding  to  0.70ju  have  small  relative 
weight. 

Elliptic  Elements  for  Selenium  Crystal  at  60°  Incidence 


Wave- 
length 
(Microns) 

Crystal  Axis  Parallel 
to  Incidence  Plane 

Crystal  Axis  Perpendicular 
to  Incidence  Plane 

A 

* 

A 

* 

0.45 

23°       11' 

35°       22' 

29°       31' 

24°       25' 

.50 

11         52 

34        14 

27        54 

23        15 

.55 

4        26 

32        27 

20        41 

24        26 

.65 

13        54 

34        12 

21        49 

27          7 

.70 

11        24 

31        13 

14        38 

26         7 

The  data  refer  to  the  light  vibrations  as  they  would  appear 
to  an  observer  stationed  just  behind,  or  within,  the  reflecting 
surface  of  the  crystal.  The  plane-polarized  incident  light  vibrating 
as  represented  in  Fig.  11  A,  the  reflected  light  vibrates  elliptically 
as  in  B.  But  as  the  latter  is  viewed  through  the  crystelliptom- 
eter,  the  observer,  facing  the  oncoming  reflected  beam,  it  would  of 
course  appear  reversed,  as  in  C.  The  phase  difference  V  of  Sec. 
IV,  which  is  what  the  crystellipometer  analysis  gives,  applies  to 
Fig.  1 1 C ;  while  the  A  given  in  the  tables  below,  corresponding  to 
Fig.  11B,  is  simply  V  minus  180°.  >J>  is  the  azimuth  angle,  whose 

a 

tangent  is  the  amplitude  ratio  -  or  r  of  Sec.  IV. 

0 


90 


LE  ROY  D.  WELD  [J.O.S.A.  &  R.S.I.,  VI 

Two  Incidences  at  Wave-length  0 . 5  Micron 


Incidence 

Axis  Parallel 

Axis  Perpendicular 

A 

* 

A 

^ 

45° 
60° 

2°       11' 
11         52 

44°        52' 
34          14 

14°         1' 

27        54 

37°        46' 
23          15 

The  values  given  in  the  second  table  are  capable,  according 
to  Dr.  R.  P.  Baker,  who  has  recently  investigated  the  application 
of  Drude's  theory  to  hexagonal  crystals,  of  yielding  the  two  sets 
of  optical  constants  of  selenium  corresponding  to  this  wave-length. 
It  is  expected  that  this  calculation  will  appear  in  a  subsequent 
paper  along  with  data  from  the  further  experimental  work  now 
in  progress. 


A.  (De$'fes) 


•A  jar  Tellurium 


A  JOY  5  e/ en  I'M  m 


30  40          SO          «•          TO          SO 

Inclination  (Dec 


ion  (De^rets) 


{p  for  Tellurium 


for  Selenium 


20  30          «0  SO  60  TO  80  >« 

Inclination 


Jn  el  in*1io 


Fig.  12 

Variations  of  A  and  S^  with  inclination  of  crystal,  to  plane  of  incidence,  for  selenium  and  tellurium 

The  above  results  refer  only  to  the  two  principal  positions  of  the 
crystal,  namely,  with  the  axis  parallel  to  and  perpendicular  to  the 


Jan.,  1922]  THE  CRYSTELLIPTOMETER  91 

incidence  plane.  Incidentally  it  was  thought  worth  while  to  study 
the  variations  of  the  elliptic  elements  with  the  angle  of  inclination 
as  the  crystal  is  turned  from  one  position  to  the  other.  Typical 
results  are  depicted  in  Fig.  12  with  selenium  and  tellurium.  No 
explanation  is  immediately  apparent  for  the  maxima  and  minima 
occurring  in  both  cases  in  the  value  of  A,  but  it  is  hoped  that  the 
mathematical  theory  may  ultimately  yield  one.  Further  exper- 
iments of  this  kind  would  be  desirable. 

In  conclusion,  the  writer  wishes  to  pay  tribute  to  the  "team 
work"  among  the  various  workers  at  the  University  of  Iowa  who 
have  contributed  toward  the  progress  of  the  research,  to  one  phase 
of  which  this  paper  is  devoted.  The  generous  cooperation  of  the 
staff  of  the  Physics  Department,  and  especially  of  Dr.  Sieg,  who 
suggested  the  problem,  is  very  greatly  appreciated. 

COE  COLLEGE,  CEDAR  RAPIDS,  IOWA, 
OCTOBER,  1921. 


•  *jj^  JOXiJjO  YV  4 


54;*>. '{,33     QC313 


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